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Theory of Computing

Quantum proofs for classical theorems

Andrew Drucker, Ronald de Wolf

Published Jan 1, 2011• 35 citations
Publisher

Abstract

Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in diverse classical (non-quantum) areas, such as coding theory, communication complexity, and polynomial approximations. In this paper we survey these results and the quantum toolbox they use.

DOI
10.4086/toc.gs.2011.002

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Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in diverse classical (non-quantum) areas, such as coding theory, communication complexity, and polynomial approximations. In this paper we survey these results and the quantum toolbox they use.
The more than thirty years old issue of the information capacity of quantum communication channels was dramatically clarified during when a number of direct quantum coding theorems were discovered. To a considerable extent this progress is due to an interplay between the quantum communication theory and quantum information ideas related to more recent development in quantum computing. It is remarkable, however, that many probabilistic tools underlying the treatment of the quantum case have their roots, and in some cases direct prototypes, in the classical Shannon theory.
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The ability (or inability) to represent or approximate Boolean functions by polynomials is a central concept in complexity theory, underlying interactive and probabilistically checkable proof systems, circuit lower bounds, quantum complexity theory, and more. In this book, the authors survey what is known about a particularly natural notion of approximation by polynomials, capturing pointwise approximation over This book covers recent progress on proving approximate degree lower and upper bounds and describes some applications of the new bounds to oracle separations, quantum query and communication complexity, and circuit complexity. The authors explain how several of these advances have been unlocked by a particularly simple and elegant technique, called dual block composition, for constructing solutions to this dual linear program. They also provide concise coverage of even more recent lower bound techniques based on a new complexity measure called spectral sensitivity. Finally, they show how explicit constructions of approximating polynomials have been inspired by quantum query algorithms. This book provides a comprehensive review of the foundational and recent developments of an important topic in both classical and quantum computing. The reader has a considerable body of knowledge condensed in an accessible form to quickly understand the principles and further their own research.